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arXiv:1707.04039 · Journal of Statistical Mechanics Theory and Experiment

Random transverse field spin-glass model on the Cayley tree: phase transition between the two many-body-localized phases

Cécile Monthus

Published Dec 1, 2017• 9 citations
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Abstract

Abstract The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by real-space-renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass many-body-localized phases involves the activated exponent ψ = 1 and the correlation length exponent ν = 1 . The spin-glass-ordered cluster containing N SG spins is found to be extremely sparse with respect to the total number N of spins: its size grows only logarithmically at the critical point N S G c r i t i ∝ ln ⁡ N , and it is sub-extensive N S G ∝ N θ in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval 0 < θ < 1 .

DOI
10.1088/1742-5468/aa9f4c

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Abstract The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by real-space-renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass many-body-localized phases involves the activated exponent ψ = 1 and the correlation length exponent ν = 1 . The spin-glass-ordered cluster containing N SG spins is found to be extremely sparse with respect to the total number N of spins: its size grows only logarithmically at the critical point N S G c r i t i ∝ ln ⁡ N , and it is sub-extensive N S G ∝ N θ in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval 0 < θ < 1 .
Strong disorder renormalization for the random transverse field Ising model leads to a complicated topology of surviving clusters as soon as d > 1. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can be obtained. Here we introduce a modified procedure called 'boundary strong disorder renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows one to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling J and random transverse fields with support [ h min , h max ]. We find the following picture: (i) for J > h max , only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster; (ii) for J < h min , only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a directed polymer model in a random medium; (iii) for h min < J < h max , both sites and bonds can be decimated, and the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified directed polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types: (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent ν typ = 1; (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears, with a correlation length exponent ν ≃ 0.75.
We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance and a uniform transverse field h . The ground state is studied via real-space renormalization to characterize the spin-glass-paramagnetic zero temperature quantum phase transition as a function of the control parameter h . In the spin-glass phase , the typical renormalized coupling grows with the length scale L as the power-law with the classical droplet exponent , where the stiffness modulus vanishes at criticality , whereas the typical renormalized transverse field decays exponentially in terms of the diverging correlation length . At the critical point , the typical renormalized coupling and the typical renormalized transverse field display the same power-law behavior L − z with a finite dynamical exponent z . The RG rules are applied numerically to chains containing L = 2 12 = 4096 spins in order to measure these critical exponents for various values of σ in the region .
The transverse-field Ising models with random exchange interactions in finite dimensions are investigated by means of a real-space renormalization-group method. The scheme yields the exact values of the critical point and critical exponent $\ensuremath{\nu}$ in one dimension and some previous results in the case of random ferromagnetic interactions are reproduced in two and three dimensions. We apply the scheme to spin glasses in transverse fields in two and three dimensions, which have not been analyzed very extensively. The phase diagrams and the critical exponent $\ensuremath{\nu}$ are obtained and evidence for the existence of an infinite-randomness fixed point in these models is found.
We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.
For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.
We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the integral is analyzed with the aid of several methods such as the semiclassical method and the gauge transformation. As an application we consider the Sherrington-Kirkpatrick model in a transverse field. Using the Landau expansion and its improved versions, we give a detailed analysis of the imaginary-time dependence of the order parameters. Integrating out the quantum part of the order parameters, we obtain the effective renormalized free energy written in terms of the classically defined order parameters. Our method allows us to obtain the spin glass-paramagnetic phase transition point $\ensuremath{\Gamma}∕J\ensuremath{\sim}1.62$ at T=0T=0T=0.